Adaptive control has been widely studied for control of nonlinear plants with modeling uncertainties. Wide ranging applications of adaptive control can be found, including control of robotics arms, flight vehicle control, and control of medical processes. Many of these approaches rely on the popular Model Reference Adaptive Control (MRAC) architecture which guarantees that the controlled states track the output of an appropriately chosen reference model. Most MRAC methods achieve this tracking by using a parameterized model of the uncertainty, often referred to as the adaptive element and its parameters referred to as adaptive weights. In MRAC, the adaptive law is designed to update the parameters in the direction of maximum reduction of the instantaneous tracking error cost (e.g., v(t)=eT(t)e(t)). While this approach ensures that the parameters take on values such that the uncertainty is instantaneously suppressed, this approach does not guarantee the convergence of the parameters to their ideal values unless the system states are persistently exciting (PE), meaning that the system states persistently receive new data.
It has been shown that the condition on PE system states can be related to a PE reference input by noting the following: If an exogenous reference input t(t) contains as many spectral lines as the number of unknown parameters, then the system states are PE, and the parameter error converges exponentially to 0. However, this condition on persistent excitation of the reference input is restrictive and often infeasible to monitor online, i.e., in real time. For example and not limitation, in flight control applications, PE reference inputs may cause nuisance, waste fuel, and cause undue stress on the aircraft. Furthermore, since the exogenous reference inputs for many online applications are event based and not known a-priori, it is often impossible to monitor online whether a signal is PE. Consequently, parameter convergence is often not guaranteed in practice for many adaptive control applications.
Various methods have been developed to guarantee robustness and efficient uncertainty suppression in adaptive control without PE reference inputs. These methods, however, are not concerned with weight convergence, but rather with instantaneously suppressing the uncertainty.